Abstract

The mode coupling among guided modes in a photorefractive crystal waveguide is investigated. We derive the coupled equations of guided modes in a photorefractive crystal waveguide. Analytical solutions of the coupled equations are obtained for a two-mode photorefractive crystal waveguide. We also numerically analyze the coupled equations of guided modes in a multimode photorefractive crystal fiber. As a result of the coupling, power may flow to the fundamental mode or the highest-order mode, depending on the direction of the crystal optical axis. In addition, we report an analytical solution of the modal power in core and cladding for all guided HEνm and EHνm modes for cylindrical fibers made of uniaxial crystals.

© 2000 Optical Society of America

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References

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  1. G. Meltz, W. W. Morsey, and W. H. Gleen, “Formation of Bragg grating in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  3. H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fibre,” Electron. Lett. 25, 797–799 (1989).
    [CrossRef]
  4. C. Shi and T. Okoshi, “Analysis of fiber-optic LP01↔LP02 mode converter,” Opt. Lett. 17, 719–721 (1992).
    [CrossRef] [PubMed]
  5. L. Solymar, “Theory of volume holographic formation in photorefractive crystal,” in Electro-Optic and Photorefractive Materials, P. Gunter, ed. (Springer-Verlag, Berlin, 1987), Chap. 3, pp. 75–97.
  6. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 4, pp. 118–182.
  7. A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. MTT-30, 790–794 (1982).
    [CrossRef]
  8. J. D. Dai and C. K. Jen, “Analysis of cladded uniaxial single-crystal fibers,” J. Opt. Soc. Am. A 8, 2021–2025 (1991).
    [CrossRef]
  9. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 7, pp. 173–177.
  11. G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
    [CrossRef]
  12. L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
    [CrossRef]
  13. P. Yeh, “Reconfigurable optical interconnection,” in Integrated Optoelectronics for Communication and Processing, C. S. Hong, ed., Proc. SPIE 1582, 3–13 (1991).
    [CrossRef]

1996 (1)

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

1992 (1)

1991 (2)

P. Yeh, “Reconfigurable optical interconnection,” in Integrated Optoelectronics for Communication and Processing, C. S. Hong, ed., Proc. SPIE 1582, 3–13 (1991).
[CrossRef]

J. D. Dai and C. K. Jen, “Analysis of cladded uniaxial single-crystal fibers,” J. Opt. Soc. Am. A 8, 2021–2025 (1991).
[CrossRef]

1990 (1)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989 (3)

H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fibre,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

G. Meltz, W. W. Morsey, and W. H. Gleen, “Formation of Bragg grating in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

1987 (1)

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

1982 (1)

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. MTT-30, 790–794 (1982).
[CrossRef]

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Chen, J. I.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Chen, Z.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Chittibabu, K. G.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Clark III, W. W.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Dai, J. D.

Gleen, W. H.

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Jen, C. K.

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Kim, B. Y.

H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fibre,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Kumar, J.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Li, L.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Marturunkakul, S.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Meltz, G.

Miller, M. J.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Morsey, W. W.

Neurgaonkar, R. R.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Okoshi, T.

Park, H. G.

H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fibre,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Salamo, G. J.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Sharp, E. J.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Shi, C.

Skinner, I.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Tonning, A.

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. MTT-30, 790–794 (1982).
[CrossRef]

Tripathy, S. K.

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Vineberg, K. A.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Wood, G. L.

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

Yeh, P.

P. Yeh, “Reconfigurable optical interconnection,” in Integrated Optoelectronics for Communication and Processing, C. S. Hong, ed., Proc. SPIE 1582, 3–13 (1991).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Electron. Lett. (2)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fibre,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

G. L. Wood, W. W. Clark III, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, “Broadband photorefractive properties of self-pumped phase conjugation in Ce-SBN:60,” IEEE J. Quantum Electron. QE-23, 2126–2135 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Tonning, “Circularly symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. MTT-30, 790–794 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

L. Li, K. G. Chittibabu, Z. Chen, J. I. Chen, S. Marturunkakul, J. Kumar, and S. K. Tripathy, “Photorefractive effect in a conjugated polymer based material,” Opt. Commun. 152, 257–261 (1996).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

P. Yeh, “Reconfigurable optical interconnection,” in Integrated Optoelectronics for Communication and Processing, C. S. Hong, ed., Proc. SPIE 1582, 3–13 (1991).
[CrossRef]

Other (3)

L. Solymar, “Theory of volume holographic formation in photorefractive crystal,” in Electro-Optic and Photorefractive Materials, P. Gunter, ed. (Springer-Verlag, Berlin, 1987), Chap. 3, pp. 75–97.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 4, pp. 118–182.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 7, pp. 173–177.

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Figures (3)

Fig. 1
Fig. 1

SBN space-charge field as function of the photoinduced grating wavelength for two doped densities: ND=1017 cm-3 (solid curve) and ND=1016 cm-3 (dashed curve).

Fig. 2
Fig. 2

Modal power versus propagation distance for ten-mode SBN rod with Φ=-π/2 and γ1k<0.

Fig. 3
Fig. 3

Modal power versus propagation distance for ten-mode SBN rod with Φ=+π/2 and γ1k>0.

Tables (4)

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Table 1 Cutoff Values of V Number

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Table 2 Propagation Constants of SBN Rod

Tables Icon

Table 3 fjk/r13no4 Factorsa and Grating Periods (ΛG)jk b

Tables Icon

Table 4 Coupling Constantsa

Equations (38)

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02π0Fi(r, ϕ)·Fj*(r, ϕ)rdrdϕ=δi j,
E=j=1nAj(z)Fj(r, ϕ)exp(-iβj z),
I=I0+j=1n-1k=j+1n{Aj (z)Ak*(z)Fj (r, ϕ)·Fk*(r, ϕ)×exp[-i(βj-βk)z]+c.c.},
=+j=1n-1k=j+1n(Δ)jk(Iavg)jk{Aj(z)Ak*(z)Fj (r, ϕ)·Fk*(r, ϕ)exp[-i(βj-βk)z]exp(iΦ)+c.c.},
(Iavg)jk=j=1n02π0|Ej|2rdrdϕ(Aeff)jk=j=1nAj Aj*(Aeff)jk,
(Aeff)ij
=02π0|Fi(r, ϕ)|2rd rdϕ02π0|Fj(r, ϕ)|2rd rdϕ02π0|Fi(r, ϕ)|2|Fj(r, ϕ)|2rd r dϕ
=102π0|Fi(r, ϕ)|2|Fj(r, ϕ)|2rdrdϕ.
n=n0+i=1n-1j=i+1nΔnij[exp(-iKijz)exp(iΦ)+c.c.],
Δ=-10r13r63r53r63r23r43r53r43r33E0SC,
E0SC=EdEqEd+Eq1+E0Ed21+E0Ed+Eq21/2,
Δ=-10r13 no4000r23 no4000r33ne4E0SC.
2E+ω2c20E=0,
d2dz2AjβjddzAj,j=1 ,, n,
j=1n2iβjdAj(z)dzFj exp(-iβjz)=j=1nk=j+1nω2(Δ)j k(Aeff)jk2c20 l=1nAlAl*{Fj*·FkFj Aj Aj* Ak×exp(-iβkz+iΦ)+Fj·Fk*Fk Aj Ak Ak*×exp(-iβjz-iΦ)+Fj*·FkFk Aj*Ak Ak×exp[-i(2βk-βj)z+iΦ]+Fj·Fk*Fj Aj Aj Ak* exp[i(βk-2βj)z-iΦ]}.
dAj (z)dz=k=1n(1-δjk)×ω2(Aeff)j k4ic20 βjFj*·FkFk*·(Δ)j kFj×exp[sign(j-k)iΦ]Aj Ak Ak*l=1nAl Al*-α2Aj,
Fj·Fk*Fj*·(Δ)jkFk
=02π0Fj·Fk*Fj*·(Δ)jkFkrd r dϕ.
Aj(z)=Mj (z)exp[iΨj(z)],
dMj (z)dz=k=1nsign(j-k)γj kMj (z)Mk(z)l=1nMl(z)-αMj,
j=1 ,, n,
ddzΨj(z)=k=1nσj kMk(z)l=1nMl(z),j=1 ,, n,
γj k=(1-δjk)ω2(Aeff)jk2c20βj×Fj·Fk*Fj*·(Δ)jkFksin(Φ),
σjk=(1-δjk)ω2(Aeff)jk4c20βj×Fj·Fk*Fj*·(Δ)jkFkcos(Φ).
Pco=πa220μ01/2βkκco2 k2no122β2Jν-12(κcoU)Jν2(κcoU)-Jν-2(κcoU)Jν(κcoU)+F222Jν-12(U)Jν2(U)-Jν-2(U)Jν(U)-νU2(1+F2)k2no122β2+F2,
Pcl=πa220μ01/2βkUW2κcl2k2no222β2Kν-2(κclW)Kν(κclW)-Kν-12(κclW)Kν2(κclW)+F222Kν-2(W)Kν(W)-Kν-12(W)Kν2(W)+νW2(1-F2)k2no222β2-F2,
F2=VUW2ν12UJν-1(U)Jν(U)-Jν+1(U)Jν(U)-12WKν-1(W)Kν(W)+Jν+1(W)Jν(W).
M1(z)[c0-β1M1(z)]-1/r
=M1(0)[c0-β1M1(0)]-1/r exp(-γ1 z),
M2(z)[c0-β2M2(z)]-r
=M2(0)[c0-β2M2(0)]-r exp(γ2 z),
γ1=ω2(Aeff)122c20β1F2·F1*F2*·(Δ)12F1sin(Φ),
γ2=ω2(Aeff)212c20β2F1·F2*F1*·(Δ)21F2sin(Φ),
Mjα(z)=Mj(z)exp(-α z).
Ψj(z)-Ψj(0)=(-1)jcot(Φ)2lnMj (z)Mj (0),
j=HE11orTE01.
γjk=(1-δjk)ω22c20βjfjk(E0SC)jk,
fjk=-Fj·Fk*Fj*·r13no4000r13no4000r33ne4FkdaFj·Fj*Fk*·Fkda

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